Homotopy types and geometries below 𝑆𝑝𝑒𝑐(ℤ)

After the first heuristic ideas about “the field of one element” <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and “geometry in characteristics 1” (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), were developed several general approaches to the construction of “geometries below <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p e c double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>Spec</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Spec}\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ”. Homotopy theory and the “the brave new algebra” were taking more and more important places in these developments, systematically explored by B. Toën and M. Vaquié, among others. This article contains a brief survey and some new results on <italic>counting problems</italic> in this context, including various approaches to zeta–functions and generalised scissors congruences. We introduce a notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> structures based on quasi-unipotent endomorphisms on homology. We also consider <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> structures based on the integral Bost–Connes algebra and its endomorphisms. In both cases we consider lifts of these structures, via an equivariant Euler charactetristic, to the level of Grothendieck rings and further lifts, via the formalism of assembler categories, to homotopy theoretic spectra.

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